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The factorial function, denoted by an exclamation mark (!), is a mathematical operation that multiplies a given non-negative integer by all the positive integers less than it. For example, the factorial of 5, written as 5!, is calculated as 5 * 4 * 3 * 2 * 1, which equals 120. The factorial of 0 is defined as 1. The factorial function is not defined for negative integers.

The factorial function is fundamental in combinatorics, the branch of mathematics concerned with counting arrangements and combinations of objects. It is used to calculate the number of ways to arrange a set of distinct items (permutations) or to select a subset of items from a larger set without regard to order (combinations). For instance, the number of ways to arrange n distinct items is n!, and the number of ways to choose k items from a set of n items is given by the binomial coefficient C(n, k) = n! / (k! * (n-k)!).

Beyond combinatorics, factorials appear in various areas of mathematics, including calculus, probability theory, and number theory. They are also used in computer science, particularly in algorithms related to permutations and combinations, and in the study of sequences and series. The rapid growth of the factorial function means that factorials quickly become very large numbers, even for relatively small inputs, which has implications for computational efficiency and data representation.